WASHINGTON — The Republican Party establishment once again won a big battle, this time in Mississippi. But it’s still facing a prolonged, bitter war with tea party insurgents.

Tea party activists and rebellious conservative groups were fuming Wednesday over Sen. Thad Cochran’s narrow win Tuesday over Chris McDaniel in Mississippi’s Republican Senate runoff election.

A crucial reason for Cochran’s 50.9 percent showing was apparently turnout by non-Republicans, notably blacks, who traditionally vote Democratic.

This party line-crossing isn’t unusual in states with open primaries. But to the conservatives who’ve become increasingly influential in the Republican Party, it’s a particularly chilling development on the eve of the 2016 presidential nominating season.

“Conservatives all over this country will wonder why the GOP should be supported,” added Brent Bozell, the chairman of the conservative group ForAmerica. “They will point to Mississippi and say, ‘There is no difference between the parties.’ “

Tea party activists on Wednesday used the Cochran win to sound warnings and energize their followers. They argued that McDaniel’s 49.1 percent of the vote – as well as his defiant vow to keep fighting – showed the movement remains on the march.

REASONS FOR OPTIMISM

That attitude concerns Republican regulars. They have nightmarish recollections of Senate races in 2010 and 2012, when mainstream candidates lost to tea party upstarts who then lost general elections that had been regarded as winnable.

The party does retain two reasons for optimism.

The biggest Senate primary battles this year were almost entirely in states that Republicans are favored to win anyway.

Also encouraging to Republicans is that the split is largely over tactics, not philosophy. Establishment Republicans have bought into the tea party agenda of dramatic spending cuts and an all-out war on the Affordable Care Act. The trouble lurks in Congress and in 2016. Congressional Republicans are divided between no-compromise conservatives who want to fight now and get-what-you-can Republicans who see success as a long-term process.

OUTREACH TO DEMOCRATS

Tea party loyalists saw Cochran’s outreach to Democrats as dangerous. His supporters poured resources into a ground game over the past few days and made an effort to turn out black voters. They were evident in Cochran ads, and a flier on cars at black churches told voters that Cochran “supports public schools” and “supports all Mississippians.”

McDaniel backers, already livid at Cochran’s ties to official Washington, were furious. “If the only way the K Street wing of the GOP establishment can win is by courting Democrats to vote in GOP primaries, then we’ve already won,” said Matt Kibbe, the president of FreedomWorks for America.

Reaching to the other side is hardly a new strategy.

Ronald Reagan appealed in 1976 to conservative Democrats in Texas and swept President Gerald Ford in that state’s presidential primary. In North Carolina, unaffiliated voters may pick a Republican or Democratic ballot, and presidential candidates in New Hampshire’s first-in-the-nation primary have long courted independent voters. In 2012, about half the states’ Republican primaries were open or allowed unaffiliated voters to participate.

Moderate Democrats often realize that in open primaries in Republican-dominated states, their best bet to elect like-minded officials is to vote in the Republican primary. “That’s the way Democrats can show their power,” said David Woodard, a Republican consultant based in Clemson, South Carolina.